Spectral Stability, Dissipativity, and Contraction: Necessary and Sufficient Robustness Characterizations for Linear and Nonlinear Systems

Spectral Stability, Dissipativity, and Contraction: Necessary and Sufficient Robustness Characterizations for Linear and Nonlinear Systems

This deposit is a compact technical note that separates asymptotic spectral stability from uniform transient robustness, and then provides a unified robustness bridge from linear systems to nonlinear systems via contraction theory.

For LTI systems , the note proves a necessary-and-sufficient characterization of 2-norm transient robustness: exponential contractivity of the semigroup is equivalent to negative definiteness of the symmetric part . As a corollary, Euclidean dissipativity is equivalent to the absence of transient amplification, clarifying why eigenvalue stability alone is insufficient for robust transient behavior in non-normal dynamics.

The note further derives an explicit transient envelope bound using Lyapunov solutions , showing how transient growth limits can be expressed in terms of the conditioning of and associated decay rates.

For nonlinear systems , the note extends the robustness framing using uniform contraction in a state-dependent metric , requiring over a forward-invariant set. Under this condition it defines a finite-horizon fragility functional aggregating spectral margin proxies, energy decay, drift accumulation, and basin-thickness terms, and proves an explicit upper bound on fragility in terms of the contraction rate , a drift Lipschitz constant, time horizon , and basin-radius proxy parameters.

Together, the results provide a unified backbone: complete (necessary-and-sufficient) robustness classification in the linear 2-norm case, and a sufficient bounded-fragility guarantee for nonlinear systems under contraction, with explicit bounds suitable for audit-style robustness arguments.

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